Download Use the Product and Power Rules for Exponents

Math 35
5.1 "Exponents"
Objectives:
*
Identify bases and exponents.
*
Use the product and power rules for exponents.
*
Use the zero and the negative integer exponent rules.
*
Use the quotient rule for exponents.
*
Simplify quotients raised to negative powers.
Identify Bases and Exponents
Natural-Number Exponents:
A natural-number exponent indicates how many times its base is to be used as a factor.
For any number x and any natural number n;
:
Example 1: (Identifying bases and exponents)
Identify the base and exponent in each expression.
2
a) ( 3)
b) 32
3
c) (5x)
Use the Product and Power Rules for Exponents
Product Rule for Exponents:
For any real number x and any natural numbers m and n;
:
(To multiply exponential expressions with the same base, keep the common base and add the exponents.)
Example 2: (Product rule)
Simplify each expression.
a) 8x4 x3
WARNING!!!
b) a2 b3 a3 b4
Examples of common errors associated with the product rule:
32 34 6= 98
Power Rule for Exponents:
For any real number x and any natural numbers m and n;
23 52 6= 105
:
(To raise an exponential expression to a power, keep the base and multiply the exponents.)
Example 3: (Power rule)
Simplify each expression.
2
32
a)
b)
x2 x3
6
c)
Page: 1
x2
4
x3
2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
5.1
Power of a Product and Quotient:
For any real numbers x and y, and any natural number n;
and
where y 6= 0:
(To raise a product to a power, raise each factor of the product to that power. To raise a quotient to a power, raise the
numerator and denominator to that power.)
Example 4:
(Product and Quotient rule)
Simplify each expression.
a)
x2 y
3
x
y2
b)
4
c)
2
6x3
5y 4
Use the Zero and the Negative Integer Exponent Rules
Zero Exponent:
For any nonzero base x
:
(A nonzero base raised to the 0 power is 1.)
Example 5: (Zero exponent)
Simplify each expression.
0
a) (5x)
b) 5x0
c)
0
d) 5a0 b
(5cd)
Negative Exponents:
For any nonzero real number x and any integer n;
:
Example 6: (Negative exponents)
Simplify each expression. Write answers using positive exponents.
a) 4
3
b) x
5 3
x
c)
x
3
2
Changing from Negative to Positive Exponents:
For any nonzero real numbers x and y, and any integers m and n;
Page: 2
:
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
5.1
Example 7: (Changing from negative to positive exponents)
Simplify each expression. Write answers using positive exponents.
1
b)
a)
c 5
s
5t
2
6
Use the Quotient Rule for Exponents
Quotient Rule for Exponents:
To divide exponential expressions with the same base, keep the common base and subtract the exponents.
For any nonzero number x and any integers m and n;
Example 8: (Quotient rule)
Simplify each expression. Write answers using positive exponents.
a5
x4 x3
a) 3
b)
a
x 5
c)
x2 y 3
7xy 4
Simplify Quotients Raised to Negative Powers
Negative Exponents and Reciprocals:
A fraction raised to a power is equal to the reciprocal of the fraction raised to the opposite power.
For any nonzero real numbers x and y, and any integer n,
Example 9: (Negative exponents and reciprocals)
Simplify each expression. Write answers using positive exponents.
3
4
y2
2
b)
a)
3
x3
Page: 3
c)
a 2 b3
a2 a3 b4
3
Notes by Bibiana Lopez